Discrete Mathematics - Sets Overview

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Questions and Answers

What defines a set in mathematics?

A collection of objects with a defined order.

An unordered collection of objects. (correct)

A collection where elements can repeat.

An ordered collection of objects.

Which notation indicates that an element is not a member of a set?

𝑒 ⊂ 𝑆

𝑒 ∉ 𝑆 (correct)

𝑒 = 𝑆

𝑒 ∈ 𝑆

Which of the following represents an open interval?

(𝑎, 𝑏) (correct)

(𝑎, 𝑏]

[𝑎, 𝑏]

[𝑎, 𝑏)

Two sets A and B are considered equal if:

All elements of A are in B and vice versa. Signup and view all the answers

What is the representation of the empty set?

∅ Signup and view all the answers

Which set correctly represents the odd positive integers less than 10?

{1, 3, 5, 7, 9} Signup and view all the answers

Which of the following correctly represents a closed interval?

[𝑎, 𝑏] Signup and view all the answers

What is the cardinality of the set 𝑆 = {𝑎, 𝑏, 𝑐, 𝑑}?

4 Signup and view all the answers

Which of the following sets is an example of an empty set?

{ } Signup and view all the answers

If 𝐴 = {1, 2, 3, {2,3}, 9}, what is the cardinality of set 𝐴?

5 Signup and view all the answers

The set of positive integers is categorized as what type of set?

Infinite set Signup and view all the answers

What notation is used to indicate that set 𝐴 is a subset of set 𝐵?

𝐴 ⊆ 𝐵 Signup and view all the answers

When can two sets A and B be considered equal?

If A is a subset of B and B is a subset of A Signup and view all the answers

How is a proper subset different from a regular subset?

A proper subset is a subset that cannot be equal to the compared set. Signup and view all the answers

What is the cardinality of the empty set?

0 Signup and view all the answers

Which of the following statements is true about finite and infinite sets?

An infinite set is characterized by the absence of a maximum element. Signup and view all the answers

What does the notation $A \subset B$ represent?

Every element of set A is an element of set B, and B contains at least one element not in A. Signup and view all the answers

How many elements are in the power set of the set $S = {1, 2, 3}$?

8 Signup and view all the answers

Which of the following sets is a power set of $S = {a, b}$?

{\emptyset, {a}, {b}, {a, b}} Signup and view all the answers

What is the Cartesian product $A \times B$ if $A = {1, 2}$ and $B = {a, b, c}$?

{(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)} Signup and view all the answers

Which statement about the element 3 in the set ${1, 2, 3, 4, 7}$ is true?

3 is an element of the set. Signup and view all the answers

How many total subsets does the set $B = {0, 3, 5, 7, 9}$ have?

16 Signup and view all the answers

If 3 is an element of the set ${1, 2, 1, 3}$, what can we conclude?

The element 3 is present regardless of the duplication rule. Signup and view all the answers

What is the universal set in the context of sets A, B, and C defined as above?

The set containing all elements from A, B, and C. Signup and view all the answers

What would the set $A \cap C$ yield if $A = {1, 2, 3, 4, 7}$ and $C = {1, 2}$?

{1, 2} Signup and view all the answers

What is the intersection of the sets {1, 3, 5} and {1, 2, 3}?

{1, 3} Signup and view all the answers

Which of the following statements is true about disjoint sets?

Their intersection is the empty set. Signup and view all the answers

What does the difference of sets A and B, denoted by A - B, represent?

Elements found in A but not in B. Signup and view all the answers

How is the complement of a set A defined in relation to the universal set U?

It includes elements in U that are not in A. Signup and view all the answers

Which of the following correctly evaluates the difference A - B for A = {1, 3, 5} and B = {1, 2, 3}?

{5} Signup and view all the answers

What is the result of the Cartesian product A × B if A = {1, 2} and B = {a, b, c}?

{(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)} Signup and view all the answers

If A = {1, 3, 5} and B = {1, 2, 3}, what is A ∪ B?

{1, 2, 3, 5} Signup and view all the answers

What does the intersection of sets A and B, denoted as A ∩ B, signify?

Elements that are common to both sets A and B Signup and view all the answers

How many elements are in the Cartesian product A × B if A contains 2 elements and B contains 3 elements?

6 Signup and view all the answers

What is the result of B × A if A = {1, 2} and B = {a, b, c}?

{(a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2)} Signup and view all the answers

If the set A = {1, 2, 3} and the set B = {3, 4, 5}, what is A ∩ B?

{3} Signup and view all the answers

Which of the following describes the concept of union of two sets?

Combining only the unique elements from both sets excluding duplicates Signup and view all the answers

In the context of the Cartesian products, what does the notation A1 × A2 × ... × An represent?

Set of ordered n-tuples derived from the sets Signup and view all the answers

If A = {1, 2} and B = {a}, what is A × B?

{(1, a), (2, a)} Signup and view all the answers

What is the primary distinction between union and intersection of sets?

Union combines unique elements, whereas intersection only includes shared elements. Signup and view all the answers

Study Notes

Discrete Mathematics - Basic Structures

Sets are unordered collections of objects
The objects in a set are called elements or members
Sets are denoted using curly brackets {}
Set notation uses ∈ to denote that an object is an element of a set, and ∉ to denote that an object is not an element
Examples of sets:
- Set O of odd positive integers less than 10 = {1, 3, 5, 7, 9}
- Set of positive integers less than 100 = {1, 2, 3, ..., 99}
Set builder notation: another way to define a set
- Example: O = {x|x is an odd positive integer less than 10}
Examples of standard sets:
- N = {0, 1, 2,...} (natural numbers)
- Z = {..., -2, -1, 0, 1, 2...} (integers)
- Z+ = {1, 2, 3,...} (positive integers)
- Q = {p/q | p ∈ Z, q ∈ Z, and q ≠ 0} (rational numbers)
- R = (real numbers)
- R+ (positive real numbers)
- C (complex numbers)
Interval Notation - defines ranges of real numbers
- Closed interval [a, b] = {x | a ≤ x ≤ b}
- Open interval (a, b) = {x | a < x < b}
- Half-open intervals:
 - [a, b) = {x | a ≤ x < b}
 - (a, b] = {x | a < x ≤ b}
Equal Sets: if A and B are equal if and only if ∀x (x ∈ A ↔ x ∈ B)
Empty Set: denoted by {} or Ø; has no elements
Cardinality: the number of distinct elements in a set, denoted by |S|
Example 1 demonstrates set notation and cardinality
- S = {a, b, c, d} |S| = 4
- A = {1, 2, 3, 7, 9} |A|= 5
- Ø = {} |Ø| = 0
Example2 demonstrates set notation and cardinality
Infinite Sets: sets that are not finite
- Z+ = {1, 2, 3...} (positive integers) is an infinite set

Subsets

Subset (⊆): Set A is a subset of set B if every element of A is also an element of B (written as A ⊆ B).
A ⊆ B ⇔ ∀ x(x ∈ A → x ∈ B)
Proper Subset (⊂): If A is a subset of B, but A ≠ B, then A is a proper subset of B (written as A ⊂ B). A proper subset cannot equal the original set.

Set Operations

Union (∪): The union of sets A and B (A∪B) is the set containing all elements that are in either A, B, or both.
Intersection (∩): The intersection of A and B (A∩B) is the set containing all elements in both A and B.
Disjoint Sets: Two sets are disjoint if their intersection is the empty set (A∩B = Ø).
Sets Difference: A − B = {x | x ∈ A and x ∉ B}
This gives the elements in A that are not in B.
Complement (Ā): The complement of set A (Ā) in universal set U is the set of elements in U that are not in A.
Generalized Unions: Using the notation ⋃i=1nAi\bigcup_{i=1}^{n}{A_i}⋃i=1nAi, to denote the union of several sets.
Generalized Intersections: Using the notation ⋂i=1nAi\bigcap_{i=1}^{n}{A_i}⋂i=1nAi, to denote the intersection of several sets.

Cartesian Product

The Cartesian product of sets A and B (A x B), is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B
Example: Let A = {1, 2} and B = {a, b, c}. A x B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}
The cardinality of the Cartesian product |A x B| = |A| * |B| = 2 * 3 = 6.
A × B × C etc...

Set Identities

These are fundamental properties of sets. Some identities include identity laws, domination laws, idempotent laws, complementation laws, commutative laws, and associative laws, etc. A table of these identities is provided(see file).
Examples illustrate how to utilize set identities and notations.

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Description

Explore the fundamentals of sets in discrete mathematics. Understand how to define sets using notation, including examples of standard sets and set builder notation. This quiz will help reinforce your knowledge of basic structures in set theory.